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Density and current of a dissipative Schrödinger operator

Hans-Christoph Kaiser, Hagen Neidhardt, Joachim Rehberg

Published Jan 1, 2002• 0 citations
Publisher

Abstract

A net current flow through an open 1-dimensional Schrödinger-Poisson system is modeled by replacing self-adjoint boundary conditions by dissipative ones. To give a rigorous definition of carrier and current densities the well-known dilation theory of dissipative operators is used where the self-adjoint dilation is regarded as the Hamiltonian of a larger closed system which contains the open one. The carrier density turns out to be performed by the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. A rigorous setup of a dissipative Schrödinger-Poisson system is outlined.

DOI
10.20347/wias.preprint.728

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A net current flow through an open 1-dimensional Schrödinger-Poisson system is modeled by replacing self-adjoint boundary conditions by dissipative ones. To give a rigorous definition of carrier and current densities the well-known dilation theory of dissipative operators is used where the self-adjoint dilation is regarded as the Hamiltonian of a larger closed system which contains the open one. The carrier density turns out to be performed by the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. A rigorous setup of a dissipative Schrödinger-Poisson system is outlined.
We regard a current flow through an open one-dimensional quantum system which is determined by a dissipative Schrödinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrödinger’s equation. This dissipative Schrödinger operator can be regarded as a pseudo-Hamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (self-adjoint) quasi-Hamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrödinger–Poisson system is outlined.
Non-self-adjoint operators play an important role in the modeling of open quantum systems. We consider a one-dimensional Schrödinger-type operator of the form −(1/2)(d/dx)(1/m)(d/dx)+V−∑κjδ(⋅−xj), Im(κj)>0, with dissipative boundary conditions. An explicit description of the characteristic function, the minimal dilation and the generalized eigenfunctions of the dilation is given. The quantities of carrier and current densities are rigorously defined. Furthermore, we will show that the current is not constant and that the variation of the current depend essentially on the chosen density matrix and the imaginary parts of the delta potentials, i.e., Im(κj). This correspondence can be used to model a recombination-generation rate in the open quantum system.
Quantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator ADA_DAD​ in a Hilbert space $\sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation K~\widetilde KK of ADA_DAD​ can be regarded as the Hamiltonian of a closed system which contains the open system $\{A_D,\sH\}$, but since K~\widetilde KK is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family {A(μ)}\{A(μ)\}{A(μ)} of maximal dissipative operators depending on energy μμμ, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.
Time-dependent [current]-density functional theory for open quantum systems (OQS) has emerged as a formalism that can incorporate dissipative effects in the dynamics of many-body quantum systems. Here, we review and clarify some formal aspects of these theories that have been recently questioned in the literature. In particular, we provide theoretical support for the following conclusions: (1) contrary to what we and others had stated before, within the master equation framework, there is in fact a one-to-one mapping between vector potentials and current densities for fixed initial state, particle-particle interaction, and memory kernel; (2) regardless of the first conclusion, all of our recently suggested Kohn-Sham (KS) schemes to reproduce the current and particle densities of the original OQS, and in particular, the use of a KS closed driven system, remains formally valid; (3) the Lindblad master equation maintains the positivity of the density matrix regardless of the time-dependence of the Hamiltonian or the dissipation operators; (4) within the stochastic Schrödinger equation picture, a one-to-one mapping from stochastic vector potential to stochastic current density for individual trajectories has not been proven so far, except in the case where the vector potential is the same for every member of the ensemble, in which case, it reduces to the Lindblad master equation picture; (5) master equations may violate certain desired properties of the density matrix, such as positivity, but they remain as one of the most useful constructs to study OQS when the environment is not easily incorporated explicitly in the calculation. The conclusions support our previous work as formally rigorous, offer new insights into it, and provide a common ground to discuss related theories.
Schrödinger equation with given, a priori known current is formulated. A nonzero current density is maintained in the quantum system via a subsidiary condition imposed by vector, local Lagrange multiplier. Constrained minimization of the total energy on the manifold of an arbitrary current density topology results into a nonlinear self-consistent Schrödinger equation. The applications to electronic transport in two-terminal molecular devices are developed and new macroscopic definition of a molecular current–voltage characteristic is proposed. The Landauer formula for the conductance of an ideal one-dimensional lead is obtained within the approach. The method is examined by modeling of current carrying states of one-dimensional harmonic oscillator.
We present systematic construction of probability and probability current densities operators for one-band single particle Pauli equations starting from the operators in Dirac electron model within Second Quantized Approach. These operators are of importance as in probability interpretation of experimental data, so in establishing of boundary conditions. It is shown that derived operators differ significally from their convential Schrodinger-type counterparts. The generalization of continuity equation for probability density under external perturbations and physical meaning of additional source terms is discussed. The presented approach can be useful in analysis of carriers dynamics described within generic multicomponent k⃗⋅p⃗\vec{k} \cdot \vec{p}k⋅p​ Hamiltonians in Envelope Function Approximation (EFA).